1) Does the web interface allow searching the database entries or is it limited to data entry?
Yes, you can perform searches
2) If I make the database with filemaker pro and upload it to the hosting service, is the number of simultaneous users still limited to something like 5 like in the "instant web publish" service?
Unless your hosting service does something very wrong, they should be hosting your files with FileMaker server advanced and this permits many more simultaneous users.
3) Can I just modify the database structure (add/delete fields etc) on my computer and sync the new structure to the hosting service pretty easily by the press of a few buttons?
4) Is the local "admin" copy on my computer kept in sync with the copy on the hosting provider; i.e. can I use the filemaker pro software on my computer to add data to the database seamlessly or should I use the web access like the rest of the team for routine data entering?
Answering 3 and 4 together: Generally, you would use remote access via your copy of FileMaker to access the hosted copy of the database. This permits you to make design changes directly to the database and also to perform operations that can't be performed through a web browser. You do need to be careful to make design changes--especially to the table definitions at times when no one is entering data into the database as this can briefly lock records and keep data from being entered. I don't know if the user will get an error message on a web published database in that situation or not.
5) Is web interaction with the database significantly more limited than using a local copy? I'm especially interested whether the users can use the web interface to add pictures to picture fields by using some kind of "browse file" dialog.
Yes, it is much more limited. See FileMaker's documentation (There are PDF files on the subject of IWP you can find and download via their downloads page that you will need to read). Uploading pictures via a browser aren't something I've tried myself, but suspect this will be a problem with an Instant Web Published database.
I'd research the documentation on web publishing FileMaker carefully and then also the cost of using a volume license install option to put FileMaker on everyone's machine, and possibly a copy of FileMaker Server (server advanced shouldn't be needed). While more expensive up front, it enables you to make full use of the native FileMaker interface that you are familiar with rather than adjusting your design to work within the limitations of access via a Web Browser.